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Chaos on a Strange Attractor

After an initial transient, the solution settles into an irregular oscillation that persists as t — °°, but never repeats exactly. The motion is aperiodic. [Pg.318]

The trajectory appears to cross itself repeatedly, but that s just an artifact of projecting the three-dimensional trajectory onto a two-dimensional plane. In three dimensions no self-intersections occur. [Pg.319]

Today this infinite complex of surfaces would be called a fractal. It is a set of points with zero volume but infinite surface area. In fact, numerical experiments suggest that it has a dimension of about 2.05 (See Example 11.5.1.) The amazing geometric properties of fractals and strange attractors will be discussed in detail in Chapters 11 and 12. But first we want to examine chaos a bit more closely. [Pg.320]

The motion on the attractor exhibits sensitive dependence on initial conditions. This means that two trajectories starting very close together will rapidly diverge from each other, and thereafter have totally different futures. Color Plate 2 vividly illustrates this divergence by plotting the evolution of a small red blob of 10,000 nearby initial conditions. The blob eventually spreads over the whole attractor. Hence nearby trajectories can end up anywhere on the attractor The practical implication is that long-term prediction becomes impossible in a system like this, where small uncertainties are amplified enormously fast. [Pg.320]

Now watch how 5(z) grows. In numerical studies of the Lorenz attractor, one finds that [Pg.321]


So far we have concentrated on the particular parameter values <7 = 10, b =, r = 28, as in Lorenz (1963). What happens if we change the parameters It s like a walk through the jungle—one can find exotic limit cycles tied in knots, pairs of limit cycles linked through each other, intermittent chaos, noisy periodicity, as well as strange attractors (Sparrow 1982, Jackson 1990). You should do some exploring on your own, perhaps starting with some of the exercises. [Pg.330]

Even when (1) has no strange attractors, it can still exhibit complicated dynamics (Moon and Li 1985). For i nstance, consider a regime in which two or more stable limit cycles coexist. Then, as shown in the next example, there can be transient chaos before the system settles down. Furthermore the choice of final state depends sensitively on initial conditions (Grebogi et al. 1983b). [Pg.446]

In the case of non-deterministic chaos (noisy) oscillations, the attractor is found to be as shown in Fig. 12.9(b) corresponding to the time series generated from thiophenol + bromate oscillator [12], which is quite different as compared to the strange attractor. The phase-plane plots of thio-phenol oscillator based on experimental results Fig. 12.9(a) displays an unusual attractor since noise in oscillations is produced by the heterogeneous reaction. [Pg.232]

In quite a few cases, chaos can be predicted based on appropriate mathematical model (Lorenz or Rosselor) (Chapter 12). The strange attractor obtained in such cases signifies Deterministic Chaos . Such models exhibit cross-catalysis and inhibition. Experimentally observed deterministic chaos can be easily ascertained using definite criteria, although prediction of complex time order is quite difficult in many cases. In terms of causality principle, the complexity in the system arises due to complex interdependence of a number of causes and effects. [Pg.317]

The significance of higher degeneracies (starting from codimension three) in the linear part is that the effective normal forms become three-dimensional, and may, as a result, exhibit complex dynamics, the so-called instant chaos, even in the normal form itself. Such examples include the normal forms for a bifurcation of an equilibrium state with a triplet of zero characteristic exponents, and a complete or incomplete Jordan block, in which there may be a spiral strange attractor [18], or a Lorenz attractor [129], respectively (the latter case requires an additional symmetry). Since we will focus our considerations only on simple dynamics, we do not include these topics in this book. [Pg.11]

In the Lorenz model, the saddle value is positive for the parameter values corresponding to the homoclinic butterfly. Therefore, upon splitting the two symmetric homoclinic loops outward, a saddle periodic orbit is born from each loop. Furthermore, the stable manifold of one of the periodic orbits intersects transversely the unstable manifold of the other one, and vice versa. The occurrence of such an intersection leads, in turn, to the existence of a hyperbolic limit set containing transverse homoclinic orbits, infinitely many saddle periodic orbits and so on [1]. In the case of a homoclinic butterfly without symmetry there is also a region in the parameter space for which such a rough limit set exists [1, 141, 149]. However, since this limit set is unstable, it cannot be directly associated with the strange attractor — a mathematical image of dynamical chaos in the Lorenz equation. [Pg.383]


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